Do men like fat or thin lips?

Full lips for women. Assuming you're talking about what kind of lips men like to see on women...Full lips convey a sense of youth., which is why lip augmentation procedures are so popular. Women with thin lips due to aging or their natural appearance can have improvement with an easy office filler injection. Male lip augmentation can be done, but must be conservative to retain natural appearance.
A balanced lip. Men like natural looking lips with the proper balance between upper and lower lips. An overly fat upper lip is unatractive and fake looking. Upper lip should be slightly narrower than lower lip but both in natural proportions.
It is individual lik. Like many other thing in human body, it is individual liking. While women on an average likes" full lips", which is a sign of beauty, most men are quite happy with their lips, whatever god gives them. In my practice, I enlage lip in women rather than in men, but I reduce lip in men, particularly african male, in selected cases as they wish.